prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Compute the smallest positive integer $M$ such that there exists a positive integer $n$ for which: - $M$ is the sum of the squares of some $n$ consecutive positive integers. - $2M$ is the sum of the squares of some $2n$ consecutive positive integers. | 4250 |
A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct 3-letter words can be made from a subset of the letters \(c, o, m, b, o\), where each letter in the list is used no more than the number of times it appears? | 33 |
What is the least positive integer $n$ such that $9n$ is a perfect square and $12n$ is a perfect cube? | 144 |
Twenty-five of the numbers $1, 2, \ldots, 50$ are chosen. Twenty-five of the numbers $51, 52, \ldots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers. | 2525 |
The 42 points $P_1, P_2, \ldots, P_{42}$ lie on a straight line in that order, such that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1 \leq n \leq 41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.) | 861 |
A sequence of real numbers $\{a_n\}_{n = 1}^\infty$ with $n=1,2,...$ has the following property: \[ 6a_n + 5a_{n-2} = 20 + 11a_{n-1} \quad (\text{for } n \geq 3). \] The first two elements are $a_1=0$ and $a_2=1$. Find the integer closest to $a_{2011}$. | 40086 |
There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\... | 5 |
The cubic polynomials $p(x)$ and $q(x)$ satisfy the following conditions: 1. $p(1) = q(2)$ 2. $p(3) = q(4)$ 3. $p(5) = q(6)$ 4. $p(7) = q(8) + 13$ Find $p(9) - q(10)$. | 52 |
Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $75\%$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $25\%$. He decides not to make it rain today. Find the smallest ... | 9 |
How many positive integers less than 2015 have exactly 9 positive integer divisors? | 13 |
For how many integers $k$ does the following system of equations have a solution other than $a=b=c=0$ in the set of real numbers? \[ \begin{cases} a^2 + b^2 = kc(a+b), \\ b^2 + c^2 = ka(b+c), \\ c^2 + a^2 = kb(c+a). \end{cases} \] | 2 |
Let $a$ be a positive real number such that $\frac{a^2}{a^4-a^2+1}=\frac{4}{37}$. Then $\frac{a^3}{a^6-a^3+1}=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 259 |
Find the remainder when \( \prod_{n=3}^{33} \left( 2n^4 - 25n^3 + 33n^2 \right) \) is divided by \( 2019 \). | 0 |
Andrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was $47$. If Andrea is $29$ years older than Jim, what is the sum of their ages now? | 79 |
How many positive integers $a$ with $a \le 154$ are there such that the coefficient of $x^a$ in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero? | 60 |
On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$, consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. ... | 67 |
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \[ (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \] with $P, Q, R \in \mathcal{A}$. Find... | 4 |
Suppose there exist constants $A$, $B$, $C$, and $D$ such that \[n^4 = A\binom{n}{4} + B\binom{n}{3} + C\binom{n}{2} + D\binom{n}{1}\] holds true for all positive integers $n \geq 4$. What is $A + B + C + D$? | 75 |
An event occurs periodically over $x$ consecutive years, followed by a break of $y$ consecutive years. The event occurred in the years 1964, 1986, 1996, and 2008, and did not occur in 1976, 1993, 2006, or 2013. Determine the first year in which the event will occur again. | 2018 |
Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\). | 19 |
Benji has a $2 \times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He kee... | 5 |
Let $a$, $b$, and $c$ be the 3 roots of $x^3 - x + 1 = 0$. Find $\frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1}$. | -2 |
Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively. | 6 |
Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$? | 640 |
By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property? | 12 |
How many ways are there to fill in a $3 \times 3$ grid of cells with $0$'s and $2$'s, one number in each cell, such that each $2 \times 2$ contiguous subgrid contains exactly three $2$'s and one $0$? | 8 |
There is a right triangle $\triangle ABC$ where $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ such that $\angle ACX = \angle XCY = \angle YCZ = \angle ZCB$ and $BZ = 2AX$. The smallest angle of $\triangle ABC$ is $\frac{a}{b}$ degrees, where $a$ and $b$ are positive integers such... | 277 |
Let $ABC$ be a triangle with $\angle A = 60^\circ$. Line $\ell$ intersects segments $AB$ and $AC$ and splits triangle $ABC$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $BX$ and $CY$ are perpendicular to $\ell$. Given that $AB = 20$ and $AC = 22$, compute $XY$. | 21 |
Given positive real numbers $a$, $b$, and $c$ such that $a^2 + b^2 = c^2$ and $ab = c$. Determine the value of the expression: \[ \left| \frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2} \right| \] | 4 |
A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the work over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked $5$ time... | 75 |
In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$. Given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$. | 260 |
Among $100$ points in the plane, with no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two p... | 1037 |
PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, what is the probability that the 4th customer orders a previously unordered item? Express the probability as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m... | 189 |
Evaluate the following expression: \[ \frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx} \] | 2005 |
What is the greatest common factor of $12345678987654321$ and $12345654321$? | 12321 |
A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the sum of the largest and smallest po... | 120 |
In triangle $ABC$, $AB = 8$, $AC = 9$, and $BC = 10$. Let $M$ be the midpoint of $BC$. Circle $\omega_1$ with area $A_1$ passes through $A$, $B$, and $C$. Circle $\omega_2$ with area $A_2$ passes through $A$, $B$, and $M$. Then $\frac{A_1}{A_2} = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute ... | 16295 |
The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$. | 60 |
For a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the number of solutions, in positive integers $n$, to the inequality $\log_4 n \leq p(n)$. | 13 |
Let $(a_1, a_2, a_3, a_4, a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of \[ \sum_{i=1}^{5} |a_i - i| = |a_1 - 1| + |a_2 - 2| + |a_3 - 3| + |a_4 - 4| + |a_5 - 5|. \] | 8 |
If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$. | 15 |
Calculate $1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 + 9 + \cdots - 96 + 97 + 98 + 99 + 100$. | 202 |
It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam? | 10 |
Find the smallest positive integer $n$ such that if $n = p_1^{a_1} p_2^{a_2} \ldots p_s^{a_s}$, then \[ \phi(n) = n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \ldots \left(1 - \frac{1}{p_s}\right) = \frac{2^5}{47}n. \] | 59455 |
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x \leq 29$, a subset of $A$ sums to $x$? | 4 |
Six chairs are arranged in a row, and six people randomly choose a seat among these chairs. Each person then decides one of three positions for their legs: 1. Feet on the floor 2. Legs crossed to the right 3. Legs crossed to the left A problem arises only if two adjacent people have the following leg positions: - The p... | 1106 |
Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that: - $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$; - For any integers $a \ge b \ge 0$, we have $g(a+b) \le f(a) + f(b)$. Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all such pairs of func... | 7993 |
Let $\phi$ be the Euler totient function. Let $\phi^k (n) = (\underbrace{\phi \circ ... \circ \phi}_{k})(n)$ be $\phi$ composed with itself $k$ times. Define $\theta (n) = \min \{k \in \mathbb{N} \mid \phi^k (n)=1 \}$. For example: - $\phi^1 (13) = \phi(13) = 12$ - $\phi^2 (13) = \phi (\phi (13)) = 4$ - $\phi^3 (13) = ... | 6037 |
Let $f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $f(f(n) + f(m)) = m + n$ for all positive integers $n, m$. Find the possible value for $f(1988)$. | 1988 |
100 schoolchildren participated in the Mathematical Olympiad where 4 tasks were given. The first problem was solved by 90 students, the second by 80, the third by 70, and the fourth by 60. No student solved all the problems. Students who solved both the third and fourth questions received an award. How many students we... | 30 |
The unit price of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed using the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to make only one purchase, what is the maximum possible number of screws he can buy? | 252 |
A sequence $\{a_n\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \ldots, 2020$ and for $i > 2020$, $a_i$ is the average of the previous 2020 terms. What is the largest integer less than or equal to $\displaystyle\lim_{n \to \infty} a_n$? | 1347 |
What are the last two digits of $2003^{2005^{2007^{2009}}}$, where $a^{b^c}$ means $a^{(b^c)}$? | 43 |
The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed. | 666 |
Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1 \pmod{2017}$. | 30576 |
Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \leq n < b$ such that $n^2$ and $n$ end in the same digit in base $b$. | 540 |
If $x$, $y$, $z$ are real numbers satisfying: \[ \begin{align*} (x + 1)(y + 1)(z + 1) &= 3, \\ (x + 2)(y + 2)(z + 2) &= -2, \\ (x + 3)(y + 3)(z + 3) &= -1, \end{align*} \] find the value of \[ (x + 20)(y + 20)(z + 20). \] | 6748 |
Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256. | 93 |
Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt{2}$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci. | 30 |
Let $A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3)$. Compute $2^A$. | 72 |
\( \triangle ABC \) has side lengths \( AB=20 \), \( BC=15 \), and \( CA=7 \). Let the altitudes of \( \triangle ABC \) be \( AD \), \( BE \), and \( CF \). What is the distance between the orthocenter (intersection of the altitudes) of \( \triangle ABC \) and the incenter of \( \triangle DEF \)? | 15 |
A box of strawberries, containing $12$ strawberries total, costs $2$. A box of blueberries, containing $48$ blueberries total, costs $3$. Suppose that for $12$, Sareen can either buy $m$ strawberries total or $n$ blueberries total. Find $n - m$. | 120 |
Determine the number of integer pairs \((x, y)\) such that \((|x| - 2)^2 + (|y| - 2)^2 < 5\). | 48 |
For any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 arbitrary finite sets. Let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A)$, $T \in P(P(A))$, $S \in T$, and $S \subseteq T$. Note that by convention, a set may never contain itself.... | 872 |
If $p$, $q$, and $r$ are nonzero integers satisfying $p^2+q^2 = r^2$, compute the smallest possible value of $(p+q+r)^2$. | 4 |
We are given a $5 \times 5$ square grid, divided into $1 \times 1$ tiles. Two tiles are called **linked** if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture, the gray tiles are the ones linked to the red tile. ; $C_3$ has center $A$ and diameter $2k$. We regard $k$ as fixed. Consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain... | 1 |
In the diagram, the circle has a radius of $\sqrt{7}$ and center $O$. Points $A$, $B$, and $C$ are on the circle. If $\angle BOC = 120^\circ$ and $AC = AB + 1$, determine the length of $AB$. | 4 |
Find the total number of integer solutions of the equation $x^5 - y^2 = 4$. | 0 |
Consider the polynomials: \[ P(x) = (x + \sqrt{2})(x^2 - 2x + 2) \] \[ Q(x) = (x - \sqrt{2})(x^2 + 2x + 2) \] \[ R(x) = (x^2 + 2)(x^8 + 16) \] Find the coefficient of \( x^4 \) in \( P(x) \cdot Q(x) \cdot R(x) \). | 0 |
While taking the SAT, you become distracted by your own answer sheet. Because you are not bound to the College Board's limiting rules, you realize that there are actually $32$ ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could ma... | 2013 |
Alice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After $35$ moves, what is the probability that she is on either vertex $A$ or $C$? | 0 |
There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. I... | 27 |
There are $5$ vertices labeled $1, 2, 3, 4, 5$. For any two pairs of vertices $u, v$, the edge $uv$ is drawn with probability $\frac{1}{2}$. If the probability that the resulting graph is a tree is given by $\frac{p}{q}$ where $p$ and $q$ are coprime, then find the value of $q^{1/10} + p$. | 127 |
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e., using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral... | 8 |
Let the rational number $\frac{p}{q}$ be closest to but not equal to $\frac{22}{7}$ among all rational numbers with denominator $< 100$. What is the value of $p - 3q$? | 14 |
Let $x, y, z \in \mathbb{R}$ and the equation $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$ holds. Find the minimum value of $x^2 + y^2 + z^2$, which can be expressed as $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $\gcd(a, b) = 1$. Determine the value of $a + b$. | 7 |
In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest side has length $12$. What is the length of the altitude (height) of the triangle that is perpendicular to the longest side? | 3 |
For real numbers $x$, let $$f(x) = \frac{4^x}{25^{x+1}} + \frac{5^x}{2^{x+1}}.$$ Then, $$f\left(\frac{1}{1 - \log_{10} 4}\right) = \frac{m}{n},$$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 15001 |
Find the remainder when $712!$ is divided by $719$. | 718 |
Suppose the side lengths of triangle $ABC$ are the roots of the polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius? | 10 |
For each positive integer $n$, let $S_n$ denote the set of positive integers $k$ such that $n^k - 1$ is divisible by $2006$. Define the function $P(n)$ by the rule: $$P(n) = \begin{cases} \min(s)_{s \in S_n} & \text{if } S_n \neq \emptyset, \\ 0 & \text{otherwise}. \end{cases}$$ Let $d$ be the least upper bound of $\{P... | 912 |
How many reorderings of \(2, 3, 4, 5, 6\) have the property that every pair of adjacent numbers are relatively prime? | 4 |
For some positive integers $m > n$, the quantities $a = \text{lcm}(m, n)$ and $b = \gcd(m, n)$ satisfy $a = 30b$. If $m - n$ divides $a$, then what is the value of $\frac{m + n}{b}$? | 11 |
If $x^3 - 3\sqrt{3} x^2 + 9x - 3\sqrt{3} - 64 = 0$, find the value of $x^6 - 8x^5 + 13x^4 - 5x^3 + 49x^2 - 137x + 2015$. | 1898 |
Suppose $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$. Compute the number of possible integer values of $n$. | 47 |
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1, 2, 3, \dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers. | 401 |
When $4 \cos \theta - 3 \sin \theta = \frac{13}{3},$ it follows that $7 \cos 2\theta - 24 \sin 2\theta = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | 122 |
A sequence $\{a_n\}$ satisfies the following conditions: - $a_0 = 0$ - $a_1 = a_2 = 1$ - $a_{3n} = a_n$ - $a_{3n+1} = a_{3n+2} = a_n + 1$ for all $n \geq 1$. Compute $a_{2021}$. | 6 |
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 1... | 250 |
Let $f(x) = x^3 + 3x - 1$ have roots $a, b, c$. Given that \(\frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + a^3}\) can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\gcd(m, n) = 1\), find \(100m + n\). | 3989 |
The George Washington Bridge is $2016$ meters long. Sally is standing on the bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\frac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge? | 1016060 |
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$. | 4648 |
There are $2022$ stones on a table. At the start of the game, Teacher Tseng will choose a positive integer $m$ and let Ming and LTF play a game. LTF is the first to move, and he can remove at most $m$ stones on his round. Then the two players take turns removing stones. Each round, they must remove at least one stone, ... | 1 |
Let $a$, $b$, $c$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$. The cubic polynomial $Q(x)$ is monic and has distinct roots $bc-a^2$, $ca-b^2$, $ab-c^2$. What is the sum of the coefficients of $Q$? | 2015000 |
Find the number of solutions to the equation: $$6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018.$$ With $\{x\}$, we denote the fractional part of the number $x$. | 5 |
Alfred and Bonnie play a game where they take turns tossing a fair coin. The winner of the game is the first person to obtain a head. Alfred and Bonnie play this game multiple times with the rule that the loser of a game goes first in the next game. Suppose Alfred goes first in the first game, and the probability that ... | 93 |
One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $\frac{1}{7}$ of the total amount of milk and $\frac{2}{17}$ of the total amount of coffee. How many people are there in Manjul’s family? | 8 |
Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up at 23 m/s with his tail. If Bob has a density of 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his center of mass plunges once he lands, assuming his center of mass was at water level when he's la... | 3 |
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